Roughly speaking, it limits the number of mathematical structures used to describe space and time that are put in place "by hand".
So background independence can be seen as extending the mathematical objects that should be predicted from theory to include not just the parameters, but also geometrical structures.
"[1] In general relativity, background independence is identified with the property that the metric of spacetime is the solution of a dynamical equation.
It is analogous and closely related to requiring in differential geometry that equations be written in a form that is independent of the choice of charts and coordinate embeddings.
The inability to make classical mechanics manifestly Lorentz-invariant does not reflect a lack of imagination on the part of the theorist, but rather a physical feature of the theory.
This and other similar results lead physicists to believe that any consistent quantum theory of gravity should include topology change as a dynamical process.
Another approach is the conjectured, but yet unproven AdS/CFT duality, which is believed to provide a full, non-perturbative definition of string theory in spacetimes with anti-de Sitter asymptotics.